What does it mean "derivative of an image"?
The term Derivative of an Image in the context you mention has two meanings.


A matrix, image, or floating point number that is derived from an image via convolution, passing the image through a two dimensional NN, the application of an FFT analysis, or some other process.  In this context, the word Derivative implies the direction of calculation: Image B is derived from image A.
A matrix or cube that represents the rate of change at in the image being processed.  The change being measured between only two adjacent pixels in a single dimension and only one direction at a time, but the applications of this technique is very limited, and such a sequence is of differences, not at all reasonable approximations of the derivative of light.  What is more useful in real recognition systems are two dimensional or hexagonal windowing (Gausian, Hamming, Hanning, trapazoidal, cosine, ...) across space and, for video, through time.  The calculus term derivative should always reference the theoretical surface being approximated using these techniques, not the discrete matrix or cube that approximates the surface.


Such multidimensional convolution and neural network based approaches are less sensitive to capture noise and orientation nuances.  Two dimensional whole image or windowed FFT techniques have met with much success because filtering the expected frequency range of features to be detected is merely an attenuation process.  Two and three dimensional splines can also be tuned to be useful in the detection of features in an orientation independent way.

In addition to gray scale analysis, color and transparency channels can be selected for independent or parallel analysis or added to the dimension of the fitting model from which the derivative is taken.

Advances in deep networks have blossomed into a new area of image processing and recognition research, bringing new hope to robotics, automated transportation, and cybernetics in general.